Finite Group Databasebeta

← #26246203

Subgroups — #26246203

Order 12032.

902 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245869C211yesmaximal, characteristic
2#384×D47C211yesmaximal, characteristic
3#20244952C211yesmaximal, characteristic, complemented
4#20244951C211yesmaximal, characteristic, complemented
5#20244950C211yesmaximal, characteristic, complemented
6#20244949C212yesmaximal, complemented
7#20244948C212yesmaximal, complemented
8#20243106C212yesmaximal
9#20243106C212yesmaximal
10#20243080C211yesmaximal, characteristic
11C47×#2100C211yesmaximal, characteristic, complemented
12C22×#10625324C2212yes—
13C2×#13093982C2211yescharacteristic
14C2×#13093982C2211yescharacteristic
15C2×C4⋊C4×D47—21no—
16C2×C4⋊C4×D47—21no—
17C2×C4⋊C4×D47C2212yes—
18C2×C42×D47C2211yescharacteristic
19#15689022C2211yescharacteristic
20#15689021C2211yescharacteristic
21#15689019C2211yescharacteristic
22#15688997—21no—
23#15688994—21no—
24#15688993C2211yescharacteristic
25#15688992—21no—
26#15688992—21no—
27#15688992C2212yes—
28C4⋊C4×Dic47—21no—
29#15688974—21no—
30#15688886C2212yes—
31#15688886C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35#15688885C2212yes—
36#15688878C2211yescharacteristic
37#15688877C2212yes—
38C47×#389C2211yescharacteristic
39C47×#384C2211yescharacteristic
40C47×#340C2212yes—
41C47×#340C2212yes—
42C47×#336C2211yescharacteristic
43C22×C4×D47—22no—
44C22×C4×D47D411yescharacteristic
45C22×C4×D47D411yescharacteristic
46C22×C4×D47C2311yescharacteristic
47C2×#10625324—24no—
48C2×#10625324—24no—
49C2×#10625324—24no—
50C22×C47⋊C8C2312yes—
51#13093982—22no—
52#13093982—24no—
53#13093982Q814yes—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47C2312yes—
58C2×C4⋊Dic47C2311yescharacteristic
59C2×C4⋊Dic47C2311yescharacteristic
60C2×Dic47⋊C4—21no—
61C2×Dic47⋊C4—21no—
62C2×Dic47⋊C4—22no—
63C2×Dic47⋊C4—22no—
64C2×Dic47⋊C4—22no—
65C4⋊C4×D47—28no—
66C4⋊C4×D47—44no—
67C4⋊C4×D47—44no—
68C2×(C2×D47)⋊C4—21no—
69C2×(C2×D47)⋊C4—21no—
70C2×C4×Dic47—21no—
71C2×C4×Dic47—21no—
72C2×C4×Dic47D411yescharacteristic
73C2×C4×Dic47D411yescharacteristic
74C2×C4×Dic47C2311yescharacteristic
75C42×D47—24no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C47⋊#95—21no—
79C22×C8×C47C2312yes—
80C2×C4⋊C4×C47—21no—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47—21no—
83C2×C4⋊C4×C47C2312yes—
84C2×C4⋊C4×C47C2311yescharacteristic
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C42×C47C2311yescharacteristic
87#95×C47—21no—
88C23×D47C2×D411yescharacteristic
89C22×C4×C47—21no—
90C22×C4×C47—22no—
91C22×C4×C47—21no—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2411yescharacteristic
95C22×Dic47—21no—
96C22×Dic47—21no—
97C22×Dic47—22no—
98C22×Dic47C2×D411yescharacteristic
99C22×Dic47C2×D411yescharacteristic
100C22×Dic47C2×D411yescharacteristic
101C2×C4×D47—22no—
102C2×C4×D47—22no—
103C2×C4×D47—22no—
104C2×C4×D47—22no—
105C2×C4×D47—22no—
106C2×C4×D47—22no—
107C2×C4×D47—44no—
108C2×C4×D47—44no—
109C2×C4×D47—44no—
110C2×C4×D47C2×Q812yes—
111C2×C4×D47#4114yes—
112C2×C4×D47#4114yes—
113C2×C4×D47#4112yes—
114C2×C8×C47—22no—
115C2×C8×C47—22no—
116C2×C8×C47—22no—
117C4⋊C4×C47—21no—
118C4⋊C4×C47—24no—
119C4⋊C4×C47—22no—
120C4⋊C4×C47—42no—
121C4⋊C4×C47—42no—
122C4⋊C4×C47—42no—
123C4⋊C4×C47C2×Q812yes—
124C42×C47—22no—
125(C2×D47)⋊C4—24no—
126(C2×D47)⋊C4—42no—
127C4⋊Dic47—21no—
128C4⋊Dic47—24no—
129C4⋊Dic47—22no—
130C4⋊Dic47—42no—
131C4⋊Dic47—42no—
132C4⋊Dic47—42no—
133C4⋊Dic47C2×Q812yes—
134Dic47⋊C4—42no—
135Dic47⋊C4—44no—
136Dic47⋊C4—44no—
137Dic47⋊C4—44no—
138Dic47⋊C4—42no—
139C4×Dic47—22no—
140C4×Dic47—24no—
141C4×Dic47—22no—
142C4×Dic47—22no—
143C4×Dic47—42no—
144C2×C47⋊C8—22no—
145C2×C47⋊C8—22no—
146C2×C47⋊C8—22no—
147#10625324—48no—
148C23×C47C22×D411yescharacteristic
149C22×D47—21no—
150C22×D47—21no—
151C22×D47—21no—
152C22×D47—21no—
153C22×D47C22⋊Q812yes—
154C22×D47#12312yes—
155C22×D47#12412yes—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—22no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—42no—
165C2×C4×C47—42no—
166C2×C4×C47—42no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47#13712yes—
170C2×C4×C47#13712yes—
171C2×C4×C47C22×Q811yescharacteristic
172C2×C4×C47C2×#4112yes—
173C2×C4×C47C2×#4112yes—
174C2×C4×C47C2×#4111yescharacteristic
175C2×Dic47—21no—
176C2×Dic47—21no—
177C2×Dic47—21no—
178C2×Dic47—21no—
179C2×Dic47—22no—
180C2×Dic47—41no—
181C2×Dic47—41no—
182C2×Dic47—41no—
183C2×Dic47—42no—
184C2×Dic47—42no—
185C2×Dic47—42no—
186C2×Dic47—41no—
187C2×Dic47—41no—
188C2×Dic47C22⋊Q812yes—
189C2×Dic47#12312yes—
190C2×Dic47#12412yes—
191C2×Dic47AGL(1,ℤ/8)12yes—
192C2×Dic47AGL(1,ℤ/8)12yes—
193C4×D47—28no—
194C4×D47—28no—
195C4×D47—88no—
196C4×D47#12518yes—
197C4×D47#12618yes—
198C8×C47—42no—
199C47⋊C8—42no—
200#23745—471nomaximal, Sylow
201C22×C47C2×C22⋊Q811yescharacteristic
202C22×C47C2×#12311yescharacteristic
203C22×C47C2×#12411yescharacteristic
204C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
205C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
206C22×C47C2×#13711yescharacteristic
207C22×C47C2×#13711yescharacteristic
208C2×D47—22no—
209C2×D47—22no—
210C2×D47—22no—
211C2×D47—22no—
212C2×D47—22no—
213C2×D47—22no—
214C2×D47#40014yes—
215C4×C47—22no—
216C4×C47—22no—
217C4×C47—42no—
218C4×C47—81no—
219C4×C47—82no—
220C4×C47C2×#12512yes—
221C4×C47C2×#12612yes—
222Dic47—22no—
223Dic47—22no—
224Dic47—42no—
225Dic47—81no—
226Dic47—82no—
227Dic47#40014yes—
228#2296—471no—
229C2×#384—471no—
230#2100—471no—
231#2100—471no—
232#2100—471no—
233#2100D4711yescharacteristic, complemented
234#2099—472no—
235#2099—472no—
236#1736—472no—
237#1736—472no—
238#1728—471no—
239C2×C47C2×#40011yescharacteristic
240C2×C47#307911yescharacteristic
241C2×C47#307911yescharacteristic
242C2×C47#309211yescharacteristic
243C2×C47#309211yescharacteristic
244C2×C47#313811yescharacteristic
245C2×C47#313911yescharacteristic
246D47—24no—
247C22×M16—472no—
248C2×#117—471no—
249C2×#117—471no—
250C22×C4⋊C4—472no—
251C22×C4⋊C4—941no—
252C22×C4⋊C4—941no—
253C22×C42—471no—
254#389—471no—
255#389—471no—
256#389—471no—
257#389C2×D4711yescharacteristic
258#385—941no—
259#385—941no—
260#384—472no—
261#384—471no—
262#384—941no—
263#384—941no—
264#384C2×D4711yescharacteristic
265C4×C4⋊C4—941no—
266C4×C4⋊C4—941no—
267#340—472no—
268#340—472no—
269#340—472no—
270#340—472no—
271#340—472no—
272#340—472no—
273#340C2×D4712yes—
274#340C2×D4712yes—
275#336—472no—
276#336—471no—
277#336C2×D4711yescharacteristic
278C47#2374511yescharacteristic, complemented, Sylow
279C23×C4—471no—
280C23×C4—471no—
281C23×C4—471no—
282C23×C4—942no—
283C2×M16—944no—
284C2×M16—944no—
285C2×M16—944no—
286C22×C8—472no—
287C22×C8C22×D4712yes—
288#117—474no—
289#117—942no—
290#117—944no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—471no—
295C2×C4⋊C4—472no—
296C2×C4⋊C4—471no—
297C2×C4⋊C4—948no—
298C2×C4⋊C4—942no—
299C2×C4⋊C4—942no—
300C2×C4⋊C4—942no—
301C2×C4⋊C4—941no—
302C2×C4⋊C4—941no—
303C2×C4⋊C4—941no—
304C2×C4⋊C4—941no—
305C2×C4⋊C4—941no—
306C2×C4⋊C4—1884no—
307C2×C4⋊C4—1884no—
308C2×C4⋊C4C22×D4712yes—
309C2×C4⋊C4C22×D4711yescharacteristic
310C2×C4⋊C4C22×D4711yescharacteristic
311C2×C22⋊C4—941no—
312C2×C22⋊C4—941no—
313C2×C42—471no—
314C2×C42—471no—
315C2×C42—471no—
316C2×C42—944no—
317C2×C42—941no—
318C2×C42—941no—
319C2×C42C22×D4711yescharacteristic
320#95—21no—
321#95—941no—
322#95—941no—
323#95—941no—
324C24—471no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—471no—
329C22×C4—472no—
330C22×C4—474no—
331C22×C4—474no—
332C22×C4—472no—
333C22×C4—471no—
334C22×C4—471no—
335C22×C4—942no—
336C22×C4—942no—
337C22×C4—942no—
338C22×C4—942no—
339C22×C4—942no—
340C22×C4—942no—
341C22×C4—941no—
342C22×C4—941no—
343C22×C4—942no—
344C22×C4—1884no—
345C22×C4—1884no—
346C22×C4—1884no—
347C22×C4D4×D4711yescharacteristic
348C22×C4D4×D4711yescharacteristic
349C22×C4C23×D4711yescharacteristic
350M16—1888no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—942no—
355C2×C8—942no—
356C2×C8—942no—
357C4⋊C4—21no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—472no—
364C4⋊C4—941no—
365C4⋊C4—944no—
366C4⋊C4—942no—
367C4⋊C4—1882no—
368C4⋊C4—1882no—
369C4⋊C4—1882no—
370C4⋊C4—1884no—
371C4⋊C4—1884no—
372C4⋊C4—1884no—
373C4⋊C4—1882no—
374C4⋊C4—1882no—
375C4⋊C4Q8×D4712yes—
376C22⋊C4—944no—
377C22⋊C4—1882no—
378C42—22no—
379C42—942no—
380C42—942no—
381C42—944no—
382C42—942no—
383C42—1882no—
384C23—472no—
385C23—472no—
386C23—472no—
387C23—941no—
388C23—941no—
389C23—941no—
390C23—941no—
391C23C2×D4×D4711yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—472no—
406C2×C4—472no—
407C2×C4—472no—
408C2×C4—478no—
409C2×C4—478no—
410C2×C4—472no—
411C2×C4—472no—
412C2×C4—948no—
413C2×C4—948no—
414C2×C4—941no—
415C2×C4—941no—
416C2×C4—941no—
417C2×C4—941no—
418C2×C4—942no—
419C2×C4—3768no—
420C2×C4—1881no—
421C2×C4—1882no—
422C2×C4—1882no—
423C2×C4—1882no—
424C2×C4—1881no—
425C2×C4—1881no—
426C2×C4—1881no—
427C2×C4—1881no—
428C2×C4#1309405312yes—
429C2×C4#1309405612yes—
430C2×C4C2×Q8×D4711yescharacteristic
431C2×C4#41×D4712yes—
432C2×C4#41×D4712yes—
433C2×C4#41×D4711yescharacteristic
434C8—42no—
435C8—1882no—
436C22—474no—
437C22—942no—
438C22—942no—
439C22—942no—
440C22—942no—
441C22—942no—
442C22—942no—
443C22C22⋊Q8×D4711yescharacteristic
444C22#123×D4711yescharacteristic
445C22#124×D4711yescharacteristic
446C22AGL(1,ℤ/8)×D4711yescharacteristic
447C22AGL(1,ℤ/8)×D4711yescharacteristic
448C22C2×#1309405311yescharacteristic
449C22C2×#1309405611yescharacteristic
450C4—22no—
451C4—22no—
452C4—42no—
453C4—82no—
454C4—81no—
455C4—474no—
456C4—942no—
457C4—942no—
458C4—3762no—
459C4—3761no—
460C4—1882no—
461C4#125×D4712yes—
462C4#126×D4712yes—
463C2—944no—
464C2#400×D4711yescharacteristic
465C2#2025320011yescharacteristic
466C2#2025320211yescharacteristic
467C2#2025331011yescharacteristic
468C2#2025331511yescharacteristic
469C2#2025376111yescharacteristic
470C2#2025376511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2100C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenter#2100D47
hypocenterC47#23745
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2100

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20244952C2semidirect
4. #20244951C2semidirect
5. #20244950C2semidirect
6. #20244949C2semidirect
7. #20244948C2semidirect
11. C47×#2100C2semidirect
233. #2100D47semidirect
278. C47#23745semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202458692
#384×D472
#202449522
#202449512
#202449502
#202449492
#202449482
#202431062
#202430802
C47×#21002
#2374547